Not Applicable
Not Applicable
1. Field of the Invention
Puzzles incorporating mazes with two dimensions of movement (left-right, forward-back) are immensely popular and relatively inexpensive to design with modern computer layout tools, but suffer from two weakness. First, with existing two-dimensional mazes, a player may view the entire maze at once, unless a structure external to the maze is used to conceal some portion. Secondly, even with three-dimensional mazes adding an xe2x80x98up-downxe2x80x99 movement, relatively simple drawing techniques allow representation using only two dimensions. Thirdly, even when a two-dimensional maze is embodied in a three-dimensional object (such as a bracelet or ring) such that the far side of the puzzle is visually blocked by the near side, the player""s mental visualization remains entirely two-dimensional, and the solution remains a two-dimensional function. A movable playing piece (flat and sliding, or spherical and rolling) often is used to move through the maze; and, to avoid falling off of the two-dimensional surface when three-dimensional movement occurs, both barriers and containing covers are used, the latter being typically transparent so as to not inhibit visibility. When the nature of the puzzle is to solve a maze, the dimensional simplicity is determined by the dimensions of movement rather than the dimensions of the maze""s construction. These weaknesses mean that a player may mentally trace out a solution using a dimensional simplification that the maze""s physical construction cannot obviate.
2. Description of the Related Art
Maze puzzles have been part of Western Civilization since Roman times. Before the American Revolution there were landscaped and tiled puzzles from Roman, Celtic, and British sources (several may be experienced within an easy half-day""s travel of Bath, England). Mosaic-solution puzzles are of more recent origin, popularly recognized as starting with the introduction of the Rubik""s Cube in the 1970""s. (However much mosaic layout problems may have presented architectural challenges to the fresco designers of the Avalon Casino on Santa Catalina, or to generations of bathroom tile layers, they were not seen as general amusements.) The 1980""s saw a revival in interest in landscape puzzles as popular amusement, leading to the construction of many such in two or even three dimensions, including among such the xe2x80x98Woozxe2x80x99 maze and amusement park located near Sacramento, Calif.
Creating a maze with two dimensions of movement can be done on a two-dimensional surface (with length l and width w) (FIG. 1) of a three-dimensional object such as a sheet of paper, plastic, or other flexible or formed material. A corresponding maze (also with length l and width w) (FIG. 2) may be drawn on the opposite side of the three-dimensional object. Each maze may be viewed completely by a view perpendicular to its surface, the l-w plane.
Topological quandaries posited by Moebius and Felix Klein (1849-1925) have inspired mathematicians and artists but otherwise had little attention paid to how they might be useful or instructive. Three-dimensional and two-dimensional puzzles have been perceived as entirely separate and distinct areas of inventive effort and gameplay.
Puzzles have been viewed as a means for both enjoyment and increasing intellectual (xe2x80x98puzzle-solvingxe2x80x99) capacities. And the topological insights of Moebius and Klein have been viewed as curious, but hardly relevant, mathematical truths. But deliberately mixing surface and dimensional complexity through design and manufacture that focuses on special topological characteristics, and doing so in a fashion that will encourage playful use as a way to foster n-dimensional visualization, is the essence of this invention. This embodiment of the invention allows even preschool youngsters the opportunity to perceive mathematical complexities usually only first encountered in high school.
A n-dimensional puzzle can be laid out and then, througha (N+1)-dimensional physical manipulation using the specific topological technique of making a unitary topological surface out of previously separated surfaces by incorporating an odd number of 180xc2x0 inversions or twists and joining the previously-separated parallel edges, be turned into an three-dimensional puzzle, whose solution requires visualization of three-dimensional movement from the start to the finish, even though the current state of the progression through the maze can always be viewed as a two-dimensional problem. This third-dimensional manipulation guarantees that at any point along the puzzle its apparantly opposing side will be hidden by that portion of the surface that is currently visible, and that accurate visualization of the dimensions of movement must include the current and final positions. The preferred embodiment of this invention is a maze puzzle in a Moebius ring, in which a moving marker piece may be used to let a player track his progress through the maze embodied in the puzzle. To prevent the moving piece from falling out during the three-dimensional manipulation by the player, the maze occupies only a portion of the flexible material""s surface and the rest is transparent and used to form an edge wall and top cover.